Some generalizations of Camina pairs and orders of elements in cosets

TL;DR

The paper develops and analyzes generalized Camina-pair conditions for finite groups, introducing the (F') relaxation of the Camina framework and proving a central equivalence: the CI condition is equivalent to the F condition for any finite group and nontrivial proper subgroup $H$. It shows that whenever (F') holds, the normal closure $N=igl\uparrow H^Gigr angle$ inherits the (F')-type property and is nilpotent, with $(G,N)$ forming a Camina pair under suitable hypotheses. The work also investigates the coset-order phenomenon via condition (O), establishing that either $O^2(H)$ is normal in $G$ with $G/O^2(H)$ a $2$-group or $H$ is solvable, and deriving consequences such as subnormality of $H$, equal-order behavior for $N_G(H)$, and related structural constraints. Collectively, these results deepen understanding of how generalized Camina conditions constrain group structure, connect to equal-order pair phenomena and Frobenius-type kernels, and illuminate implications for simple groups and related conjectures.

Abstract

In this paper, we investigate certain generalizations of Camina pairs. Let $H$ be a nontrivial proper subgroup of a finite group $G$. We first show that every nontrivial irreducible complex character of $H$ induces homogeneously to $G$ if and only if for every $x\in G\setminus H$, the element $x$ is conjugate to $xh$ for all $h\in H$. Furthermore we prove that if $xh$ is conjugate to either $x$ or $x^{-1}$ for all $h\in H$ and all $x\in G\setminus H$, then the normal closure $N$ of $H$ in $G$ also satisfies the same condition, and $N$ is nilpotent. Finally, we determine the structure of $H$ under the assumption that for every element $x\in G\setminus H$ of odd order, the coset $xH$ consists entirely of elements of odd order.

Theorems & Definitions (60)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4